From Stein to Weinstein and Back. Symplectic Geometry of Affine

BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 52, Number 3, July 2015, Pages 521–530 S 0273-0979(2015)01487-4 Article electronically published on February 19, 2015 From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds, by Kai Cieliebak and Yakov Eliashberg, AMS Colloquium Publications, vol. 59, American Mathematical Society, Providence, RI, 2012, xii+364 pp., ISBN 978- 0-8218-8533-8, hardcover, US $78. The book under review [5] is a landmark piece of work that establishes a fundamental bridge between complex geometry and symplectic geometry. It is both a research monograph of the deepest kind and a panoramic companion to the two fields. The main characters are, respectively, Stein manifolds on the complex side and Weinstein manifolds on the symplectic side. The connection between the two is established by studying the corresponding geometric structures up to homotopy, or deformation, a context in which the h-principle plays a fundamental role. Stein manifolds are fundamental objects in complex analysis of several variables and in complex geometry. There are several equivalent definitions, but the most intuitive one is the following: they are the complex manifolds that admit a proper holomorphic embedding in some Cn (they are in particular noncompact). Stein manifolds can be thought of as being a far-reaching generalization of domains of holomorphy in Cn. Other important classes of examples are noncompact Riemann surfaces and complements of hyperplane sections of smooth algebraic manifolds. The latter are actually examples of affine (algebraic) manifolds, meaning that they are described by polynomial equations in Cn. While affine manifolds are open analogues of smooth algebraic varieties, Stein manifolds form a much larger class within the analytic category. They are sometimes thought of as being natural domains for holomorphic maps with values in arbitrary complex manifolds [27]. In the wake of Cartan’s foundational theorems A and B and Serre’s fundamental GAGA principle [3, 38], much of the literature on Stein manifolds dealt with their analytic properties and used coherent sheaves as a primary tool. This well- established and rich subject is taken up in several classical monographs, for example [17,24]. One upshot is that the biholomorphism type of a Stein manifold is very rigid: for example, there are uncountably many nonbiholomorphic Stein manifolds that are C∞-small perturbations of the open ball in Cn [26]. Here we come upon the first shift in point of view operated by the book under review: Stein manifolds are not studied up to biholomorphism, but rather up to deformation, or homotopy, of the Stein structure. Such a shift in point of view may seem entirely unnatural from the perspective of complex analysis, but it is natural from the topological perspective of the Oka principle for Stein manifolds, which we now describe. As early as 1939 Oka [34] had discovered that the second Cousin problem on the existence of globally defin- ing functions for positive analytic divisors inside domains of holomorphy of Cn admits a holomorphic solution if and only if it admits a continuous solution (see also [35, pp. 24–35]). This discovery gradually evolved into one of the most powerful tools for the study of Stein spaces, called the Oka principle, formulated as follows: “On a reduced Stein space X, problems which can be cohomologically formulated 2010 Mathematics Subject Classification. Primary 53D05, 53D10, 53D99, 32E10, 32Q28, 55P10, 57R42. c 2015 American Mathematical Society 521 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 522 BOOK REVIEWS have only topological obstructions. In other words, such problems are holomorphically solvable if and only if they are continuously solvable” [17, p. 145], see also [37]. The Oka principle appears nowadays to have been a precursor for an entirely new strand of mathematics known under the name of homotopy principle, or h- principle.Theh-principle was formulated as a broad unifying concept by Gromov in his 1970 ICM address [21]. As we understand it now, it refers to a class of theorems whose content is the following: the existence of a “formal” solution of a PDE (of geometric origin and usually heavily underdetermined) implies the existence of a “genuine” solution (see below for a discussion of formal solutions vs. genuine solutions). Formal solutions are always objects of a topological nature, while genuine solutions are always objects of an analytic nature. It is often the case that the correspondence between formal solutions and genuine solutions holds for arbitrary finite-dimensional families, in which case we speak of the parametric h-principle. The parametric h-principle can be succinctly formulated as saying that the inclusion of the space of genuine solutions into the space of formal solutions is a (weak) homotopy equivalence. (The adjective “weak” here refers to the property that the inclusion induces isomorphisms on all homotopy groups, and the parenthe- ses signify that the stronger statement about homotopy equivalence always holds, though weak homotopy equivalence is the only thing that is used in practice; see the discussion in [13, §6.2].) In the case of one-dimensional families, the h-principle be- comes a statement about homotopies: the existence of a formal homotopy between two genuine solutions implies the existence of a genuine homotopy. The classical reference for the h-principle in its various incarnations is the book by Gromov [19]. A recent and friendly treatment is given by Eliashberg and Mishachev in [13]. To grasp the meaning of genuine solutions vs. formal solutions, it is instructive to examine the case of smooth immersions. This was historically the first instance of the h-principle other than Oka’s work, and it is known as the Smale–Hirsch theorem [25,40]. Though slightly weaker, the original statement proved by Smale and Hirsch is essentially equivalent to the following: the parametric h-principle holds for smooth immersions of a k-dimensional manifold M k into an n-dimensional manifold N n for n>k. In other words, the inclusion Imm(M k,Nn) → FImm(M k,Nn)of the space of immersions into the space of formal immersions, consisting of injective bundle maps from TM → M to pullbacks f ∗TN → M by arbitrary smooth maps f : M → N, is a homotopy equivalence. The inclusion associates to a smooth immersion f : M k → N n the formal immersion df : TM → f ∗TN. The particular case M = S1, N = R2 was established much earlier by Whitney and Graustein [43]. Smale proved his famous sphere eversion theorem [39] by showing that the space of formal immersions S2 → R3 is connected, a statement that reduces to the van- ishing of the second homotopy group of RP 3. This is symptomatic of the fact that formal solutions to a problem governed by the h-principle are classified by algebraic invariants. The h-principle also holds in certain holomorphic setups, most notably for holomorphic immersions of Stein manifolds into Cn (Gromov and Eliashberg [22, 23]). Using the same kind of methods, Gromov and Eliashberg proved that any Stein 3n +1 manifold of dimension n embeds holomorphically into C 2 , an optimal re- sult [12]. Underlying these developments is the classical Oka principle stated above, and we refer to the recent monograph [14] for a self-contained treatment of this cir- cle of ideas. It is worth emphasizing that, in this context, although the methods License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use BOOK REVIEWS 523 are very much of a topological flavor, the point of view is reminiscent of the classical one in that the complex structure is fixed. By contrast, deformations of Stein structures as considered in the book under review allow the complex structure to change. Let us now discuss the notion of Stein structure and that of deformation of a Stein structure. These use the following alternative definition of Stein manifolds due to Grauert [16]: A complex manifold (W, J) is Stein if and only if it admits an exhausting, i.e., proper and bounded from below, J-convex function ϕ : W → R. c c The condition of J-convexity means that the 2-form ωϕ := −dd ϕ, with d ϕ := dϕ◦ J, satisfies ωϕ(v, Jv) > 0 for all tangent vectors v =0.A Stein structure on an even- dimensional smooth manifold W is a pair (J, ϕ) consisting of a complex structure J and of a J-convex exhausting smooth function ϕ : W → R. Note that the condition of J-convexity is open in the C2-topology, so that the function ϕ can and often will be assumed to be Morse, meaning that its critical points are nondegenerate. If the function ϕ can be chosen to have only finitely many (nondegenerate) critical points, we speak of a Stein manifold of finite type.Adeformation, or homotopy, of Stein structures is a continuous one-parameter family (Jt,ϕt), t ∈ [0, 1], of generalized Morse functions subject to the additional requirement that critical points “do not escape to infinity” during the homotopy. Here by generalized Morse function we mean a function whose critical points are either nondegenerate or of birth-death type (these functions form the codimension 1 stratum in the space of all functions, the open stratum being that of Morse functions). The condition of no escape to infinity is subtle, and without imposing it the theory would be void: any two Stein structures on Cn would be homotopic! This is carefully explained in the book [5, §11.6]; see in particular Remark 11.24. In formal terms, it means that on any smalltimesubintervalof[0 , 1] the manifold W can be written as a countable union k k W = k≥1 Wt over families Wt which are smooth in the time parameter t and consist of regular sublevel sets for ϕt.

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